cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A254766 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

Original entry on oeis.org

5, 11, 9, 17, 13, 23, 21, 17, 27, 35, 23, 21, 41, 31, 29, 39, 37, 53, 33, 31, 41, 59, 39, 49, 37, 35, 43, 63, 53, 37, 49, 77, 59, 47, 75, 83, 65, 53, 73, 51, 45, 61, 71, 59, 79, 69, 95, 55, 49, 101
Offset: 1

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Author

Wolfdieter Lang, Feb 11 2015

Keywords

Comments

The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = A007522(n) = 7 + 8*A139487(n) is given in A254929(n).
The positive fundamental solutions of the first classes are given in (A254764(n), A254765(n)).
For comments and the Nagell reference see A254764.

Examples

			The first pairs [x2(n), y2(n)] of the fundamental positive solutions of the second class are (we list the prime A007522(n) as first entry): [7,[5,3]], [23,[11,7]], [31,[9,5]], [47,[17,11]], [71,[13,7]], [79,[23,15]], [103,[21,13]], [127,[17,9]], [151,[27,17]], [167,[35,23]], [191,[23,13]], [199,[21,11]], [223,[41,27]], [239,[31,19]], [263,[29,17]], [271,[39,25]], ...

Crossrefs

Cf. A007522, A139487, A254929, A254764, A254765, A254760, A254761, A254762, A254763,

Formula

a(n)^2 - 2*(A254929(n))^2 = A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.
a(n) = 3*A254764(n) - 4*A254765(n), n >= 1.

A254764 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

Original entry on oeis.org

3, 5, 7, 7, 11, 9, 11, 15, 13, 13, 17, 19, 15, 17, 19, 17, 19, 19, 23, 25, 23, 21, 25, 23, 27, 29, 29, 25, 27, 35, 31, 27, 29, 33, 29, 29, 31, 35, 31, 37, 43, 35, 33, 37, 33, 35, 33, 41, 47, 35
Offset: 1

Views

Author

Wolfdieter Lang, Feb 12 2015

Keywords

Comments

For the corresponding term y1(n) see A254765(n).
For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A254766(n) and A254929(n).
The present solutions of the first class are the smallest positive ones.
See the Nagell reference Theorem 111 p. 210 for the proof of the existence of solutions (the discriminant of this binary quadratic form is +8 hence it is an indefinite form with an infinitude of solutions if there exists at least one).
See the Nagell reference Theorem 110, p. 208 for the proof that there are only two classes of solutions for this Pell equation, because the equation is solvable, and the primes A007522(n) do not divide 4.
The present fundamental solutions are found according to the Nagell reference Theorem 108, p. 205, adapted to the case at hand, by scanning the following two inequalities for solutions x1(n) = 2*X1(n) + 1 and y1(n) = 2*Y1(n) + 1. The intervals for X1(n) and Y1(n) to be scanned are ceiling((sqrt(2+p(n))-1)/2) <= X1(n) <= floor(sqrt((2*p(n))-1)/2), with p(n) = A007522(n) and 0 <= Y1(n) <= floor((sqrt(p(n)/2)-1)/2).
The general positive proper solutions for both classes are obtained by applying positive powers of the matrix M = [[3,4],[2,3]] on the fundamental column vectors (x(n),y(m))^T.
The least positive x solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including also prime 2) are given in A002334.

Examples

			The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (we list the prime A007522(n) as first entry):
  [7, [3, 1]], [23, [5, 1]], [31, [7, 3]], [47, [7, 1]], [71, [11, 5]], [79, [9, 1]], [103, [11, 3]], [127, [15, 7]], [151, [13, 3]], [167, [13, 1]], [191, [17, 7]], [199, [19, 9]], [223, [15, 1]], ...
a(3)^2 - 2*A254765(3)^2 = 7^2 - 2*3^2 = 31 = A007522(3).

References

  • T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

Crossrefs

Cf. A007522, A254765, A254766, A254929, A254760, A254761, A254762, A254763, A002334.

Formula

a(n)^2 - 2*A254765(n)^2 = A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

A254765 Fundamental positive solution y = y1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 3, 7, 3, 1, 7, 9, 1, 5, 7, 3, 5, 1, 9, 11, 7, 1, 9, 5, 11, 13, 11, 3, 7, 17, 11, 1, 7, 13, 3, 1, 7, 13, 5, 15, 21, 11, 7, 13, 5, 9, 1, 17, 23, 1
Offset: 1

Views

Author

Wolfdieter Lang, Feb 12 2015

Keywords

Comments

For the corresponding term x1(n) see A254764(n).
See A254764 for comments and the Nagell reference.
The least positive y solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including also prime 2) are given in A002335.

Examples

			A254764(4)^2 - 2*a(4)^2 = 7^2 - 2*1^2 = 47 = A007522(4).

Crossrefs

Cf. A007522, A254764, A254766, A254929, A254760, A254761, A254762, A254763, A002335.

Formula

A254764(n)^2 - 2*a(n)^2 = A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.

A254931 Fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1, (primes congruent to 1 or 7 mod 8).

Original entry on oeis.org

3, 4, 7, 5, 8, 11, 7, 12, 15, 10, 8, 13, 16, 9, 14, 17, 23, 13, 18, 11, 27, 14, 19, 12, 22, 17, 25, 28, 23, 18, 14, 32, 35, 19, 17, 22, 30, 25, 36, 39, 16, 28, 23, 31, 21, 19, 40, 20, 18, 38
Offset: 1

Views

Author

Wolfdieter Lang, Feb 12 2015

Keywords

Comments

The corresponding terms x = x2(n) are given in A254930(n).
The y2-sequence for the second class for the primes congruent to 1 (mod 8), which are given in A007519, is 2*A254763. For the primes congruent to 7 (mod 8), given in A007522, the y2-sequence is A254929.
For comments and the Nagell reference see A254760.

Examples

			a(4) = 2*7 - 3*3 = 5.
A254930(4)^2 - 2*a(4)^2 = 9^2 - 2*5^2 = 31 = A001132(4) = A007522(3).
See A254930 for the first pairs (x2(n), y2(n)).

Crossrefs

Cf. A254930, A001132, A002334, A002335, A007519, 2*A254763, A007522, A254929, A254760.

Programs

  • Mathematica
    Reap[For[p = 2, p < 1000, p = NextPrime[p], If[MatchQ[Mod[p, 8], 1|7], rp = Reduce[x > 0 && y > 0 && x^2 - 2 y^2 == p, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; y2 = xy[[-1, 2]] // Simplify; Print[y2]; Sow[y2]]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2019 *)

Formula

A254930(n)^2 - 2*a(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer satisfying this (generalized) Pell equation.
a(n) = 2*A002334(n+1) - 3*A002335(n+1), n >= 1.
Showing 1-4 of 4 results.

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